Inequalities in One TriangleGeometry CP1 (Holt 5-5) K. Santos

Theorem 5-5-1 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Y

X Z(In a triangle, larger angle opposite longer side)

If XZ > XY then m<Y > m<Z

ExampleList the angles from largest to smallest. A 10 B

5 8 C

Largest angle: <C <A

Smallest angle: <B

Theorem 5-5-2If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. A

B C(In a triangle, longer side is opposite larger angle)

If m<A > m<B then BC > AC

ExampleList the sides from largest to smallest. A 50 60 B

C

Find the missing angle first: 180 – (50 +60) m < C = 70

Largest side:

Smallest side:

Triangle Inequality Theorem (5-5-3)The sum of the lengths of any two sides of a triangle is greater than the length of the third side. X

Y Z

XY + YZ > XZYZ + ZX > YXZX + XY > ZYadd two smallest sides together first

Examples:Can you make a triangle with the following lengths?

1. 4, 3, 64 + 3 > 6 add the two smallest numbers7 > 6 triangle2. 3, 7, 23 + 2 > 75 > 7 not a triangle3. 5, 3, 23 + 2 > 55 >5 not a triangle

Example:

The lengths of two sides of a triangle are given as 5 and 8. Find the length of the third side.

8 – 5 = 35 + 8 = 13

The third side is between 3 and 133 < s < 13 where s is the missing side

Example:

Use the lengths: a, b, c, d, and e and rank the sides from largest to smallest. d

59 Hint: find missing angles a e c 61 59 60 bIn the left triangle: e > b > aIn the right triangle: c > d > e

But notice side e is in both inequality statements. So, by using substitution property you can write one big inequality statement.

c > d > e > b > a